Poisson algebra

Summary

A Poisson algebra is a vector space with two products, an associative product and a Lie bracket. They are linked by one rule: bracketing with a fixed element is a derivation of the associative product. The main examples are algebras of smooth functions on a Poisson manifold, in particular on a symplectic manifold, with the Poisson bracket. Poisson algebras are the algebraic setting of classical Hamiltonian mechanics, and deformation quantization turns them into noncommutative associative algebras.